用实变量研究平稳高斯过程的卷绕数
Studying the winding number for stationary Gaussian processes using real variables
摘要：
本文研究了定义在直线上的平面平稳高斯过程的卷绕数。在较弱的条件下，我们得到了当时域趋于无穷大时卷绕数的渐近方差和中心极限定理。在渐近制度，我们的离散方法是等价的连续研究文献中，我们的主要结果扩展了现有的。我们的模型允许一个一般的依赖性的坐标的过程和不可微的其中之一。此外，超出我们的一般框架，我们认为作为例子的一个近似的卷绕数的过程，其坐标都是不可微的和卷绕数的过程，这是不完全稳定的。
英文摘要：
We consider the winding number of planar stationary Gaussian processes defined on the line. Under mild conditions, we obtain the asymptotic variance and the Central Limit Theorem for the winding number as the time horizon tends to infinity. In the asymptotic regime, our discrete approach is equivalent to the continuous one studied previously in the literature and our main result extends the existing ones. Our model allows for a general dependence of the coordinates of the process and non-differentiability of one of them. Furthermore, beyond our general framework, we consider as examples an approximation to the winding number of a process whose coordinates are both non-differentiable and the winding number of a process which is not exactly stationary.
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